arXiv subjects

Andres Zuniga

Publications and source records attributed to Andres Zuniga.

8 recordsLinked to original sources

arXiv2026

A sharp three-particle fractional Hardy inequality and an angular Selberg-type identity

We establish a sharp three-particle fractional Hardy inequality for the Laplacian of order $s\in(0,1)$ in dimension $d\geq 4-2s$ (Theorem 1.1), involving an explicit intrinsically three-body interaction potential $V_{s,3}$. The inequality holds with the optimal two-particle fractional Hardy constant $C_{fH}(d,s)$, which is shown to be sharp relative to the fixed potential $V_{s,3}$. This potential $V_{s,3}$ strictly dominates the standard pairwise Coulomb-type interaction and captures genuine three-body effects. As a consequence, we derive a nontrivial many-particle fractional Hardy inequality for $N\geq 3$, and, in the regime $N>d+1$, obtain an improved Coulomb-type inequality with a strictly larger constant, agreeing in spirit with the results of Hoffmann-Ostenhof et al. [M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom, "Many-particle Hardy inequalities", J. Lond. Math. Soc. 77 (2008), no. 2, 99-115] and Lundholm [D. Lundholm, "Geometric extensions of many-particle Hardy inequalities", J. Phys. A: Math. Theor. 48 (2015), no. 17, 175203]. The proof relies on a fractional ground-state representation method adapted to three-particle interactions, combined with an explicit evaluation of the resulting nonlocal interaction term. This evaluation is achieved through a new singular integral identity of Selberg-type (Theorem 1.2), extending the three-fold formula of Grafakos-Morpurgo [L. Grafakos and C. Morpurgo, "A Selberg integral formula and applications", Pacific J. Math. 191 (1999), no. 1, 85-94] beyond the radial setting. This identity provides the analytic mechanism underlying the emergence of the three-body potential and may be of independent interest in harmonic analysis.

math-ph↗

arXiv2024

Existence of solutions to a quasilinear nonlocal PDE

In this paper, we introduce a new class of quasilinear operators, which represents a nonlocal version of the operator studied by Stuart and Zhou [1], inspired by models in nonlinear optics. We will study the existence of at least one or two solutions in the cone $X=\{u\in H^s_0(Ω): u\geq 0\}$ using variational methods. For this purpose, we analyze two scenarios: the asymptotic sublinear and linear growth cases for the reaction term. Additionally, in the sublinear case, we establish a nonexistence result.

math.AP↗

arXiv2022

Symmetry breaking and weighted Euclidean logarithmic Sobolev inequalities

On the Euclidean space, we establish some Weighted Logarithmic Sobolev (WLS) inequalities. We characterize a symmetry range in which optimal functions are radially symmetric, and a symmetry breaking range. (WLS) inequalities are a limit case for a family of subcritical Caffarelli-Kohn-Nirenberg (CKN) inequalities with similar symmetry properties. A generalized carré du champ method applies not only to the optimal solution of the nonlinear elliptic Euler-Lagrange equation and proves a rigidity result as for (CKN) inequalities, but also to entropy type estimates, with the full strength of the carré du champ method in a parabolic setting. This is a significant improvement on known results for (CKN). Finally, we briefly sketch some consequences of our results for the weighted diffusion flow.

math.AP↗

arXiv2020

A nonlocal isoperimetric problem with density perimeter

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent $α$, under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter $γ$. We show that for a wide class of density functions the energy admits a minimizer for any value of $γ$. Moreover these minimizers are bounded. For monomial densities of the form $|x|^p$ we prove that when $γ$ is sufficiently small the unique minimizer is given by the ball of fixed volume. In contrast with the constant density case, here the $γ\to 0$ limit corresponds, under a suitable rescaling, to a small mass $m=|Ω|\to 0$ limit when $p d-α+1$.

math.AP↗

arXiv2019

Prescribed energy connecting orbits for gradient systems

We are concerned with conservative systems $\ddot{q}=\nabla V(q), \; q\in\mathbb{R}^N$ for a general class of potentials $V\in C^1(\mathbb{R}^N)$. Assuming that a given sublevel set $\{V\leq c\}$ splits in the disjoint union of two closed subsets $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$, for some $c\in\mathbb{R}$, we establish the existence of bounded solutions $q_c$ to the above system with energy equal to $-c$ whose trajectories connect $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of $\nabla V$ on $\partial\mathcal{V}^c_{\pm}$. Next, we illustrate applications of the existence result to double-well potentials $V$, and for potentials associated to systems of Duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions $(q_c)$.

math.DS↗

arXiv2018

Continuity of minimizers to weighted least gradient problems

We revisit the question of existence and regularity of minimizers to weighted least gradient problems on a fixed bounded domain, subject to a Dirichlet boundary condition, in the case where the boundary data is continuous and the weight function is C^2 and bounded away from zero. Under suitable geometric conditions on the domain in R^n we construct continuous solutions of the above variational problem in any dimension n>=2, by extending the Sternberg-Williams-Ziemer technique to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in a conformal metric determined by the weight function. This results complements the approach of Jerrard, Moradifam and Nachman since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities. The proof relies on an application of a strict maximum principle for sets with area-minimizing boundary established by Leon Simon.

math.AP↗

arXiv2016

On the heteroclinic connection problem for multi-well gradient systems

We revisit the existence problem of heteroclinic connections in $\mathbb{R}^N$ associated with Hamiltonian systems involving potentials $W:\mathbb{R}^N\to \mathbb{R}$ having several global minima. Under very mild assumptions on $W$ we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells, where length is computed with respect to a degenerate metric having conformal factor $\sqrt{W}.$ Then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times, it gives rise to a heteroclinic connection between the two wells. This work improves upon the approach of P.Sternberg in $\texttt{Vector-valued local minimizers of nonconvex}$ $\texttt{variational problems}$, and represents a more geometric alternative to the approaches for finding such connections described, for example, by N.D. Alikakos and G.Fusco in $\texttt{On the connection problem for potentials with}$ $\texttt{several global minima}$, by S.V. Bolotin in $\texttt{Libration motions of natural dynamical systems}$, by J. Byeon, P. Montecchiari, and P. Rabinowitz in $\texttt{A double well potential}$ $\texttt{system}$, and by P. Rabinowitz in $\texttt{Homoclinic and heteroclinic orbits for a class of Hamiltonian}$ $\texttt{systems}$.

math.AP↗

arXiv2013

A two end family of solutions for the Inhomogeneous Allen-Cahn equation in R^2

In this work we construct a family of entire bounded solution for the singulary perturbed Inhomogeneous Allen-Cahn Equation $\ep^2÷\left(a(x)\nabla u\right)-a(x)F'(u)=0$ in $\R^2$, where $\ep\to 0$. The nodal set of these solutions is close to a "nondegenerate" curve which is asymptotically two non paralell straight lines. Here $F'$ is a double-well potential and $a$ is a smooth positive function. We also provide example of curves and functions $a$ where our result applies. This work is in connection with the results found by Z.Du and B.Lai, Z.Du and C.Gui, and F. Pacard and M. Ritore, in "Transition layers for an inhomogeneus Allen-Cahn equation in Riemannian Manifolds", "Interior layers for an inhomogeneous Allen-Cahn equation", "From the constant mean curvature hypersurfaces to the gradient theory of phase transitions" respectively, where they handle the compact case.

math.AP↗